Explicit analytical solutions for liquid infiltration into capillary tubes: dynamic and constant contact angle

We derive new analytical solutions for liquid infiltration into a gas-filled capillary tube, whose inlet is connected to a liquid reservoir held at a constant pressure. We generalize the Lucas-Washburn theory to account for a model for dynamic contact angle that assumes the nonequilibrium Young force to depend linearly on the velocity of the gas-liquid interface. Like Lucas and Washburn, we neglect inertial forces. Using the Lambert function, we derive explicit analytical solutions for the interface position, velocity, and acceleration as a function of time. Consistent with previous work, which used more general models for dynamic contact angle, we can distinguish between five infiltration scenarios: horizontal infiltration, upward infiltration (capillary rise), as well as steady-state, accelerating, and decelerating downward infiltration. We determine the mutually exclusive conditions for the different infiltration scenarios to occur in terms of the nondimensional parameters that define the problem. Moreover, we develop 2D and 3D diagrams that show which parameter combination results in which infiltration scenario. Our analytical solutions are also valid in the limit where the dynamic contact angle becomes constant. For a constant contact angle, accelerating downward infiltration occurs only if the initial interface is not located at the tube inlet but further down the tube. For the special case in which the contact angle is constant, the liquid pressure at the tube inlet is equal to the gas pressure, and the interface is initially located at the tube inlet, our solution for upward infiltration is identical to a solution previously reported in the literature.